## Positive solution of a problem of Emden-Fowler type with a free boundary.(English)Zbl 0637.34013

Let us consider the question of existence of a solution of the problem: Find $$T>0$$ and $$y\in C[0,T]\cap C^ 1[0,T)\cap C^ 2(0,T)$$ such that $(1)\quad y''(t)+q(t)y(t)^{\gamma}=0,\quad t\in (0,T),\quad y(0)=y(T)=0,\quad y'(0)=\alpha,\quad y(t)>0,\quad t\in (0,T),$ where $$\gamma\geq 1$$, $$\alpha >0$$ and q, a positive function, are given. If $$q(t)=t^{\beta}$$, $$\beta$$ a real number, (1) reduces to the Emden- Fowler equation whose origin lies in theories concerning gaseous dynamics in astrophysics. Under the additional assumptions:
i) $$q\in C^ 2(0,+\infty)$$, $$\forall t>0:$$ $$q(t)>0$$, $$t\to t^{\gamma}$$, q(t) belongs to $$L^ 1(0,1).$$
ii)$$\eta$$,$$\eta$$ ”$$\in L^ 1(1,+\infty)$$, where $$\eta \in C^ 2(0,+\infty)$$ is defined by $$\eta (t)=[q(t)]^{-1/\gamma +3}$$ and $$\int^{+\infty}_{1}[\eta (t)]^{-2} dt=+\infty.$$
iii) If $$\gamma >1:$$ $\lim_{t\downarrow 0}t^{-1} \eta (t)[\int^{+\infty}_{t}\eta (s)| \eta ''(s)| ds]^{2/\gamma - 1}=0,$ the author proves that the problem (1.1) has a unique solution (T,y). If $$q(t)=t^{\beta}$$, it is shown that (1) has a solution if and only if $$\gamma +2\beta +3>0$$. Moreover a monotone iteration scheme holds for the approximation of a positive solution.